Fractal dynamics of complex systems

One of the most important properties of systems is complexity. In a simple way, wecan define the complexity of a system in terms of the number of elements that it contains,the nature and number of interrelations, and the number of levels of embeddedness.When a high level of complexity exists in a system, it is considered a complex system.Although there is no single agreed-on definition of complex systems, they share somethemes: (i) they are inherently complicated or intricate, so that they have factors such asthe number of parameters affecting the system or the rules governing interactions ofcomponents of the system; (ii) they are rarely completely deterministic, and stateparameters or measurement data may only be known in terms of probabilities; (iii)mathematical models of the system, are usually complex and involve non-linear, illposed,or chaotic behavior; and (iv) the systems are predisposed to unexpected outcomes(so-called emergent behavior). To try to understand the dynamics of these systemsdiverse mathematical tools have been developed. A new scientific discipline withgreatimpact in the analysis of the complex systems has been developed in recent years,called fractal analysis.The study of the complex systems in the framework of fractal theory has beenrecognized as a new scientific discipline, being sustained by advances that have beenmade in diverse fields ranging from physics to economics. In this chapter the basicconcepts of fractal analysis of complex systems are briefly explained and three examplesof fractal analysis are provided: epilepsy, oil crude price market volatility, and supplychain volatility in the telecom industry.